A Joint Limit Theorem for Laplace Transforms of the Riemann Zeta–Function

In the paper, a joint limit theorem in the sense of weak convergence of probability measures on the complex plane for Laplace transforms of the Riemann zetafunction is obtained

Approximation by mean of the function given by dirichlet series by absolutely convergent Dirichlet series

It is proved an uniform on compact sets approximation by mean of the general Dirichlet series

On Joint Distribution of General Dirichlet Series

In the paper a joint limit theorem in the sense of the weak convergence in the space of meromorphic functions for general Dirichlet series is proved under weaker conditions as in [1]

On Zeros of Periodic Zeta Functions

We consider zeta functions ζ(s; a) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; a)) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; a)).Розглянуто дзета-функції ζ(s; a ), що задані рядами Діріхлє з мультиплікативними періодичними коефiцiєнтами, та доведено, що для деяких класів функцій F функції F(ζ(s; a )) мають нескінченну кількість нулів у критичній смузі. Наприклад, це виконується для sin(ζ(s; a ))

Bagchi's Theorem for families of automorphic forms

We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.Comment: 15 page

МОДИФИКАЦИЯ ТЕОРЕМЫ МИШУ

The Mishou theorem asserts that a pair of analytic functions from a wide class can be approximated by shifts of the Riemann zeta and Hurwitz zeta-functions $(\zeta(s+i\tau), \zeta(s+i\tau, \alpha))$ with transcendental $\alpha$, $\tau\in\mathbb{R}$, and that the set of such $\tau$ has a positive lower density. In the paper, we prove that the above set has a positive density for all but at most countably many \varepsilon>0, where $\varepsilon$ is the accuracy of approximation. We also obtain similar results for composite functions $F(\zeta(s),\zeta(s,\alpha))$ for some classes of operator $F$.В 2007 г. Г. Мишу доказал совместную теорему унивурсальности для дзета-функции Римана $\zeta(s)$ и дзета-функции Гурвица $\zeta(s,\alpha)$ с трансцендентным параметром $\alpha$ об одновременном приближении пары функций из широкого класса аналитических функций сдвигами $(\zeta(s+i\tau), \zeta(s+i\tau,\alpha))$, $\tau\in \mathbb{R}$. Он получил, что множество таких сдвигов, приближающих данную пару аналитических функций, имеет положительную нижнюю плотность. В статье получено, что множество таких сдвигов имеет положительную плотность для всех \varepsilon>0, за исключением счетного множества значений $\varepsilon$, где $\varepsilon$ -- точность приближения.Результаты аналогичного типа также получены для сложных функций $F($ $\zeta(s),\zeta(s,\alpha))$ для некоторых классов операторов $F$ в пространстве аналитических функций

On the sign of the real part of the Riemann zeta-function

We consider the distribution of $\arg\zeta(\sigma+it)$ on fixed lines $\sigma > \frac12$, and in particular the density $$d(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\arg\zeta(\sigma+it)| > \pi/2\}|\,,$$ and the closely related density $$d_{-}(\sigma) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: \Re\zeta(\sigma+it) < 0\}|\,.$$ Using classical results of Bohr and Jessen, we obtain an explicit expression for the characteristic function $\psi_\sigma(x)$ associated with $\arg\zeta(\sigma+it)$. We give explicit expressions for $d(\sigma)$ and $d_{-}(\sigma)$ in terms of $\psi_\sigma(x)$. Finally, we give a practical algorithm for evaluating these expressions to obtain accurate numerical values of $d(\sigma)$ and $d_{-}(\sigma)$.Comment: 22 pages, 3 tables. To appear in Proceedings of the International Number Theory Conference in Memory of Alf van der Poorten (Newcastle, Australia, 2011